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Studies on structural behaviour of long-span continuous composite beams with flexible shear studs of limited ductility.

1. Introduction

The structural system of a composite beam is essentially a T beam with a thin wide concrete flange connected with a steel section. The benefits of composite action are increased strength and stiffness with 1.5-2.5x of both the moment resistances and the flexural rigidities of the steel sections, leading to considerable economy in the size of the steel beam used. Composite beams are connected to steel or composite columns through various types of connections, which provide different degrees of flexural continuity over the internal supports. Due to the hogging moment capacities at internal supports, the load-carrying capacities of the continuous and the semi-continuous composite beams are much higher than the load-carrying capacities of the simply supported beams with similar geometrical configurations and material properties.

This paper reports a comprehensive three-dimensional finite element study to examine the structural behaviour of continuous composite beams with the consideration of the flexibility of shear connectors. With the full incorporation of material, geometrical and interfacial non-linearities, the results of the proposed finite element models compare quite well with test results of continuous composite beams with a wide range of geometrical configurations, material properties, arrangements of shear connectors as well as loading and boundary conditions. A comprehensive parametric study is also conducted based on the proposed finite element model, which studies the structural behaviour of continuous composite beams with different material properties and geometrical configurations. Composite beams with both ductile and non-ductile shear connectors are included to investigate the effects of flexibility of the shear connectors on the overall structural behaviour of continuous composite beams.

2. Literature review

The design of simply supported composite beams with both solid concrete slabs and composite slabs was covered in BS5950: Part 3 (BSI 1990), ANSI/AISC 360-05 (AISC 2005), Eurocode 4 (BSI 2004), AS2327 (Standards Australia 1996), and Hong Kong Steel Code (Buildings Department 2005) based on the plastic section theory. Prescriptive design rules were provided in current codes of practice for the design of continuous composite beams with the incorporation of moment redistribution. In these design rules (BSI 1990, 2004; AISC 2005), bending moment envelopes were calculated according to elastic uncracked or cracked section analyses allowing for the maximum percentages of moment redistribution. The maximum PMR were specified according to the section classifications of the steel sections allowing for different rotation capacities of the composite sections over internal supports.

A large number of tests were conducted in the past thirty years to study the structural behaviour of simply supported and continuous composite beams with different material properties, geometrical configurations, and arrangements of shear connectors (Chapman and Balakrishnan 1964; Terazkiewicz 1967; Hamada and Longworth 1976; Hope Gill and Johnson 1976; Ansourian 1981; Jayas and Hosain 1989). Both double-span and triple-span continuous composite beams were covered in the tests. Most of the tests were conducted on continuous composite beams with solid slabs under point loads. Large number of shear connectors were provided to ensure full shear connection. The section depths of these composite beams ranged from 283 mm to 415 mm, and the span-over-depth ratios, L/ [D.sub.c], ranged from 10 to 15. The yield strengths of the steel sections were found to be smaller than 355 N/[mm.sup.2].

A number of finite element and finite difference models on the general behaviour of simply supported and continuous composite beams with flexible shear connectors were also reported by Gattesco (1999), Loh, Uy, and Brandford (2004a), Hassan and Mourad (2002), and Cas et al. (2004). In these models, both the concrete flanges and steel sections were modelled with beam-column elements and shear connectors were modelled with discrete or continuous interfacial spring elements. With the carefully selected stress-strain curves for both reinforced concrete flanges and steel sections as well as the nonlinear load-slippage curves of the shear connectors, it was demonstrated that these finite element models were able to predict the structural behaviour of both simply supported and continuous composite beams satisfactorily. However, it should be noted that the interaction between bending stresses and shear stresses were not modelled at all in ordinary beam-column elements (ABAQUS 2008). Thus, these finite element and finite difference models could only predict the structural behaviour of composite beams which failed under large bending moments. Significant error might result if these models are employed to simulate the structural behaviour of short composite beams under large shear forces. Liang et al. (2004) proposed a three-dimensional finite element model to simulate the structural behaviour of a continuous composite beam in which shell elements were adopted to model both the steel section and the concrete slab. Typically, only two shell elements were arranged in the vertical direction of the steel web to evaluate both the bending and the shear stresses in the steel section together with their interaction. Failure of the continuous composite beam was defined either by a large deformation of the steel section or numerical divergence due to brittle behaviour of the concrete. Wang and Chung (2006) proposed a comprehensive two-dimensional finite element model for the integrated analysis and design of simply supported composite beams under either sagging or hogging moment. The model is further extended to study the structural behaviour of semi-continuous composite beams by incorporating various levels of rotational rigidity and strength at middle supports (Wang 2012). Pi, Bradford, and Brian (2005) and Ranzi and Bradford (2007) provide a numerical technique for the simulation of composite beams as an alternative to using commercial finite element software. Bradford (2012) conducted a comprehensive review of the numerical analysis of composite steel-concrete beams, with a particular attention on the modelling of composite beams containing deep trapezoidal slabs cast onto profiled steel sheeting. It was concluded that while robust algorithms are available, there was considerable scope for improvement of these models as an alternative to undertaking expensive testing programs in order to formulate practical design procedures.

With the increasing usage of long-span composite beams of high-strength materials as well as profiled steel decking with deep and narrow troughs in modern composite buildings, highly non-ductile deformation characteristics of shear connectors were observed in both push-out tests and composite beam tests in the past few years (Patrick 2000; Hegger and Doinghaus 2002). In the more recent push-out tests of shear connectors embedded in composite slabs with deep troughs conducted by Popo-Ola (2003), apparent unloading in the shear connectors was observed at a slippage of 4-8 mm in most of the test specimens.

Moreover, in the series of simply supported composite beams tested by Loh, Uy, and Brandford (2004b) under hogging moments, very large slippage (6-8 mm) was required to fully mobilise the moment capacities of the composite sections under hogging moments. Hence, the ductility of shear connectors is important to the structural behaviour of continuous composite beams, especially in hogging moment regions. Thus, it is necessary to study the effects of nonlinear as well as non-ductile deformation characteristics of shear connectors on the structural behaviour of continuous composite beams.

3. Objectives and scope

The objectives of the present study are:

* To propose a comprehensive three-dimensional finite model to study the structural behavious of continuous composite beams with the incorporation of flexible shear connectors with limited ductility.

* To study the structural behaviour of continuous composite beams through a comprehensive parametric study on continuous composite beams with different material properties and geometrical configurations.

* To provide detailed understanding and to establish design guidance for the practical design of continuous composite beams with medium to large span-over-depth ratios, high-strength materials, and non-ductile shear connectors.

Continuous composite beams with both ductile and non-ductile shear connectors are analysed to investigate the effects of the flexibility of shear connectors on the overall structural behaviour of continuous composite beams. Both the load-deflection characteristics and the PMR from internal supports to mid-spans of continuous composite beams are studied, and the results are compared with the relevant design rules in Eurocode 4: Part 1.1 (BSI 2004).

4. Finite element modelling

A three-dimensional finite element model is established with the commercial finite element package ABAQUS (2008) to study the structural behaviour of continuous composite beams, and the finite element meshes of the finite element model are presented in Figure 1.

4.1. Finite element model

Concrete slabs are modelled with eight-noded solid elements, C3D8. Typically, 4 elements are arranged in the vertical direction while 10 elements are arranged in the transverse direction so that the shear lag effects can be incorporated and modelled with sufficient accuracy. Steel sections are modelled with eight-noded double-curvature shell elements, S8. Typically, 10 shell elements are arranged in the vertical direction of the webs while 8 elements are arranged in the transverse direction of the steel flanges. Through a mesh sensitivity study, it is found that the distributions and the interactions of both the direct stresses and the shear stresses in the continuous composite beams can be modelled accurately through such arrangement of the elements. Local buckling on both the compressive flanges and the webs of the steel sections under hogging moments can also be captured.

4.2. Shear connectors

Every shear connector is modelled with one horizontal spring, one transverse spring and one vertical spring in order to simulate its horizontal and transverse shear as well as pull-out actions. However, it should be noted that while 19-mm-diameter-headed shear connectors were adopted in all tests, no information on the actual load-slippage curves of the shear connectors is provided in the literature. The normalised load-slippage curve proposed by Oehlers and Sved (1995) is adopted and expressed as follows for the ease of reference:

[F.sub.h] = [P.sub.s][(1 - [e.sup.-[beta]S]).sup.[alpha]] (1)

where [alpha] is taken as 1.2 and [beta] is taken as 2.0. The shear resistance, [P.sub.s], of the shear connector is taken as 72 kN after a trial-and-error process. The load-slippage curve derived from Equation (1) is shown also in Figure 1 together with test data of a 19-mm-headed shear connector reported by Lawson (1990) for direct comparison.

4.3. Contact between steel section and concrete flange

Spring contact elements with large compressive stiffnesses but zero tensile stiffnesses are placed in the interfaces between the steel sections and the concrete flanges to prevent element intrusion. Thus, the transfer of the vertical shear forces between the steel sections and the concrete flanges can be modelled properly in both hogging and sagging moment regions.

4.4. Material models

Failure of both steel and concrete materials is defined according to the failure surfaces in Figure 2. Cracks in concrete slabs are simulated with smeared cracking model with the incorporation of both tension stiffening and tension softening. In this material model, any crack is smeared into a series of micro-cracks in the elements within the cracked region. The tensile strength of concrete is taken as 10% of its compressive strength and assumed to reduce linearly from its peak value to zero at a tensile strain of 0.1% at which the crack is fully open.

Tensile reinforcement is also modelled assuming that it is fully bonded to surrounding concrete. Smeared reinforced concrete layers are formed for reinforced concrete as shown in Figure 1. Both the compressive and the tensile strengths of smeared reinforced concrete layers are taken as the resultant values of the tensile reinforcement and the surrounding concrete at various strain levels.

4.5. Geometrical non-linearity

Geometrical non-linearity is incorporated into the finite element models, so that large deformation in the critical sections after yielding can be predicted accurately. In addition, the first eigen-mode is input as the initial imperfection in the nonlinear analysis and the magnitude of the maximum initial imperfection is taken as 25% of the web thickness of the steel sections (Liang et al. 2004). Figure 1 illustrates a typical initial imperfection.

4.6. Formation of plastic hinge

In general, the plastic hinges are regarded to be fully developed when the maximum strain at the critical composite cross section reaches the limiting value, [[epsilon].sub.max], which is defined as follows (Wang and Chung 2006; Wang 2012):

[[epsilon].sub.max] = 5.72 x [p.sub.y]/[E.sub.s] [square root of ([P.sub.y]/250)] [2]

where [p.sub.y], is the measured yield strength of the steel section; [E.sub.s], is the measured Young's modulus, and it may be taken as 205 kN/[mm.sup.2] in the absence of test data.

4.7. Calibration against physical tests

Five tests on continuous composite beams are carefully selected from the literature (Hamada and Longworth 1976; Hope Gill and Johnson 1976; Ansourian 1981) for the calibration of the proposed finite element model. Table 1 presents the geometrical configurations and loading and boundary conditions of the continuous composite beams. It should be noted that the span-over-depth ratios, L/[D.sub.c], of the composite beams range from 8.8 to 18.7 while the flange width-to-beam span ratios, [B.sub.s]/L, range from 0.18 to 0.33. Thus, a wide range of continuous composite beams with different geometrical configurations are covered in the study. Moreover, both double and triple-span continuous composite beams under various loading and boundary conditions are calibrated. The continuous composite beams typically fail in a two-stage mechanism in most cases, namely:

* Stage 1: Failure at internal support under hogging moment; and

* Stage 2: Failure at mid-span under sagging moment.

Figure 3 shows the typical failure mode of the continuous composite beams. However, it should be noted that Beam CTB1 is loaded unsymmetrically, and the first plastic hinges are formed at the mid-spans of the loaded spans of the continuous composite beams. Beam CB11 is triple-span continuous composite beams and point loads are applied only on the internal spans of the continuous composite beams. The first plastic hinges are also found to be formed at the mid-spans of the internal spans. In addition, plastic local buckling is occasionally found in the compressive flanges of the steel sections near internal supports after plastic hinges have been fully developed. The load-deflection curves and load-carrying capacities from both finite element modelling and tests are presented in Figure 4 and Table 2 respectively for direct comparison. With the provision of detailed information on material properties and arrangements of shear connectors, it is found that the results from proposed finite element model compare well with those from tests, and the proposed model is able to be adopted as an extension of studies on structural behaviour of continuous composite beams with a wider range of geometrical configurations and material properties.

5. Deformation characteristics of shear connectors

The proposed finite element model is adopted herein to study the structural behaviour of continuous composite beams with medium to large span-over-depth ratios, L/ [D.sub.c]. By incorporating different values of slippage limits in the horizontal springs, the effects of the flexibility of shear connectors can be modelled and studied accordingly. Three types of shear connectors with different nonlinear deformation characteristics are considered in the model as shown in Figure 5.

* Shear connector A: Ductile shear connector;

* Shear connector B: Non-ductile shear connector with gentle unloading path; and

* Shear connector C: Highly non-ductile shear connector with steep unloading path.

It is noted that, there are few data on the unloading paths of non-ductile shear connectors reported from tests in the literature. In order to simplify the problem, the shear connector is assumed to start to unload at the slippage limit while the shear resistance is reduced linearly to zero after 2 mm of additional slippage or to half of its original value after 5 mm of additional slippage, as shown in Figure 5. Thus, numerical divergence due to the sudden loss of the resistances in the shear connectors can be efficiently suppressed.

6. Parametric study

A total of three double-span continuous composite beams and three continuous composite beams at internal spans with different steel sections are included in the parametric study. The geometrical configurations of the continuous composite beams are shown in Figures 6 and 7. It is noted that all of the continuous composite beams are under uniformly distributed loads. The span of these continuous composite beams is taken as 10 m to simulate secondary beams commonly used in modern composite buildings. The overall depths of these composite beams range from 439 to 747 mm. The nominal yield strengths of steel sections are taken as 355 and 450 N/[mm.sup.2] in various cases. The nominal yield strength of tensile reinforcement is taken as 460 N/[mm.sup.2], and the cube strength of concrete is taken as 30 N/[mm.sup.2].

The dimensions of the composite slab adopted in the current study are also illustrated in the same figure. The width of troughs is taken as 85 mm and the height of troughs is taken as 70 mm. According to Patrick (2000) and Popo-Ola (2003), shear connectors embedded in such composite slabs are likely to exhibit non-ductile deformation characteristics.

6.1. Scheme

Table 3 summarises the details of the parametric study. A total of 10 groups of continuous composite beams are covered in this study with the following controlling parameters:

* span-over-depth ratio, L/[D.sub.c];

* ductility of shear connector;

* strength of steel section;

* degree of shear connection;

* amount of tensile reinforcement; and

* boundary condition.

6.1.1. Section depth

Different section depths are adopted in the continuous composite beams under Group A0 to study the effects of section depths and span-over-section depth ratios, L/ [D.sub.c] on the structural behaviour of continuous composite beams.

6.1.2. Ductility of shear connector

Shear connectors with different degrees of ductility are adopted in the continuous composite beams under different groups. Ductile shear connectors (Shear connectors A) are used in the continuous composite beams under Group A0, while non-ductile shear connectors (Shear connectors B and C) are used in the continuous composite beams under Groups A1 and A2. Thus, the effects of the non-ductile deformation characteristics of shear connectors can be studied.

6.1.3. Degree of shear connection

Different degrees of shear connection are adopted in the continuous composite beams under Groups A3 and A4. Ductile shear connectors are used in the composite beams under Group A3 and highly non-ductile shear connectors are used in the composite beams under Group A4. Thus, the effects of partial shear connection and non-ductile shear connectors on the moment redistribution of continuous composite beams can be investigated accordingly.

6.1.4. Amount of tensile reinforcement

Different amounts of tensile reinforcement are adopted in the hogging moment regions of the continuous composite beams under Group A5 to study the effects of flexural continuity at internal supports on the structural behaviour of continuous composite beams.

6.1.5. Strength of steel section

It should be noted that the steel strength of all the continuous composite beams under Groups A0 to A5 is taken as 355 N/[mm.sup.2]. In order to study the effects of steel strengths on the structural behaviour of continuous composite beams, the study is repeated under Groups B2 to B4 with the yield strength of the steel section taken as 450 N/[mm.sup.2].

6.1.6. Boundary condition

Since end-spans are normally considered to be more critical than internal spans in continuous composite beams, most of the continuous composite beams under the present study are double-span continuous composite beams except for the composite beams under Group C0, which are modelled as continuous composite beams at internal spans.

6.2. Basic moment capacities

Figures 8 and 9 present the sagging and the hogging moment capacities of the three composite beams with different degrees of shear connection. The nominal yield strength of steel sections is taken as 355 N/[mm.sup.2] for the composite beams in Figure 8 and 450 N/[mm.sup.2] for the composite beams in Figure 9, respectively. It is noted that composite action between the steel sections and the concrete flanges increases sagging moment capacities by an amount of 48-153%, and the hogging moment capacities by an amount of 6-31%, when compared with those of the steel sections. This demonstrates the important effects of composite action on the load-carrying capacities of the composite beams. This also implies the importance to ensure that the composite action is developed properly between the steel sections and the concrete flanges.

7. Results of parametric study

The results of the parametric study under each of the groups of continuous composite beams are presented in Figures 10-19, and the details of the results are presented in the following sections. Design guidance is also established for continuous composite beams with medium to large span-over-depth ratios, and with the possible non-ductile deformation characteristics of shear connectors. It is noted that all the continuous composite beams in the present study fail in a two-stage mechanism. In the Stage 1 failure, plastic hinges are formed at internal supports under hogging moments, and in the Stage 2 failure, plastic hinges are formed at mid-spans under sagging moments.

7.1. Different section depths (Groups A0, A1 and A2)

Steel sections of different depths are adopted in the continuous composite beams under Group A0 with the adoption of ductile shear connectors, namely, Shear connectors A. Both the load-deflection curves and the development of sagging and hogging moments of the continuous composite beams under this group are presented in Figure 10. The occurrence of first crack in the concrete flanges in the hogging moment regions is also marked in the same graphs for direct comparison. It is noted that cracks in concrete flanges are assumed to be fully open at a limiting strain failure criteria of 0.1% according to the concrete material model shown in Figure 2. It is found that the first crack typically occurs at a load level about 35% of the ultimate load-carrying capacities in order to mobilise the hogging moment capacities of the continuous composite beams.

7.1.1. Effects of section depths

Table 4 presents the PMR of the continuous composite beams. It should be noted, that by varying the span-over-depth ratios, L/[D.sub.c], from 13.3 to 22.8, the PMR increase from 32 to 49%. This implies the important effects of L/[D.sub.c] ratios on the PMR of continuous composite beams. A higher L/[D.sub.c] ratio normally leads to a larger percentage of moment redistribution.

7.1.2. Effects non-ductile shear connectors

Non-ductile shear connectors with gentle unloading path, namely, Shear connectors B are used in the continuous composite beams under Group A1 while highly non-ductile shear connectors, namely, Shear connectors C are used in the continuous composite beams under Group A2. The results of the analyses are presented in Figures 11 and 12 for the continuous composite beams under Groups A1 and A2, respectively. For the continuous composite beams with section depths larger than or equal to 747 mm (Beams EB1.355-Bp-N1 and EB1.355-Cp-N1), premature knock-down is found due to the occurrence of the non-ductile deformation characteristics of shear connectors before the formation of the second plastic hinges. A premature knock-down is defined to be a sudden deduction in the overall load-carrying capacity over a relative short deflection history (Wang 2010a, 2010b). This reduces the ultimate load-carrying capacities of the continuous composite beams. However, for the continuous composite beams with section depths smaller than 747 mm, the non-ductile deformation characteristics of shear connectors occur well after the formation of the second plastic hinges. It should be noted that for composite beams with deep sections, larger strains in the extreme fibres in both the steel sections and the concrete flanges are induced in mobilising the cross-sectional moment capacities. This leads to large slippage at the steel-concrete interfaces, which are equal to the differences in direct strains between the upper surfaces of steel sections and the bottom surfaces of concrete flanges. This can be clearly observed from Table 4, in which the maximum slippage of the shear connectors in the continuous composite beams at both Stage 1 and 2 failures is presented. Thus, continuous composite beams with large section depths (larger than or equal to 747 mm) are more prone to suffering from the non-ductile deformation characteristics of shear connectors than those with smaller steel sections. It should be noted that the unloading is much gentler for the continuous composite beams under Group A1 than those under Group A2 because of the different deformation characteristics of the shear connectors.

In addition, the load-deflection curves of the bare steel beams of the three continuous composite beams covered in this study are also plotted in Figure 12(a) for direct comparison. It is noted that, after the occurrence of the non-ductile deformation characteristics of shear connectors, the load-carrying capacities of the continuous composite beams approach the load-carrying capacities of the bare steel sections without any composite action. It is also noted that non-ductile deformation characteristics of shear connectors lead to 16-23% reduction in the sagging moment capacities, and about 5% reduction in the hogging moment capacities of the composite beams. Thus, the ultimate load-carrying capacities of the continuous composite beams with the occurrence of non-ductile deformation characteristics of shear connectors covered in this study are reduced by 12-20%.

7.2. Different arrangements of shear connectors (Groups A3 and A4)

7.2.1. Effects of different degrees of shear connection

Ductile shear connectors are arranged in different configurations in the continuous composite beams under Groups A3. There are 2 shear connectors provided in every trough in Beam EB2.355-Ap-N1 while there is only 1 shear connector provided in every alternate trough of Beam EB2.355-Aa-N1. The degrees of shear connection of the continuous composite beams range from 0.12 to 0.49 in the sagging moment regions and, 0.92-4.32 in the hogging moment regions. The load-deflection curves of these continuous composite beams are presented in Figure 13(a). It is found that the ultimate load-carrying capacities of the continuous composite beams with different degrees of shear connection differ by about 22%. A higher degree of shear connection normally leads to a higher ultimate load-carrying capacity.

Figure 13(b) presents the development of bending moments in both the hogging and the sagging moment regions. The PMR range from 41% for the continuous composite beam with the highest degree of shear connection at 0.49-33% for the continuous composite beam with the lowest degree of shear connection at 0.12. It should be noted that full shear connection in the hogging moment regions can be easily achieved in practice because of the usual provision of tensile reinforcement. In general, composite beams with a higher degree of shear connection can attract more bending moments in the sagging moment regions after the first plastic hinges are formed in the hogging moment regions. This, in turn, leads to higher PMR.

7.2.2. Effects of non-ductile shear connectors

The analyses are repeated in the continuous composite beams under Group A4 with the adoption of highly non-ductile shear connectors, namely, Shear connectors C. The load-deflection curves and the development of bending moments of the continuous composite beams are presented in Figure 14. It is noted that non-ductile deformation characteristics of shear connectors happens right after the formation of the second plastic hinge in Beam EB2.355-Cp-N1, in which the degree of shear connection is 0.49 in the sagging moment region. However, for the other two continuous composite beams with lower degrees of shear connection (Beams EB2.355-Ce-N1 and EB2.355-Ca-N1), premature knockdown occurs due to the occurrence of the non-ductile deformation characteristics of the shear connectors and with a sudden deduction in the overall load-carrying capacity over a relative short deflection history, before the formation of the second plastic hinges at mid-spans.

This reduces the ultimate load-carrying capacities of the continuous composite beams. It is found that the occurrence of non-ductile deformation characteristics of shear connectors reduces the load-carrying capacities of the continuous composite beams by 15-25%. It should also be noted that lower degrees of shear connection normally leads to larger slippage in the shear connectors as can be found through the observation of the maximum slippage of the continuous composite beams. Thus, in general, continuous composite beams with lower degrees of shear connection (less than 0.5 in sagging moment regions) are more prone to suffering from the premature knock-down due to non-ductile shear connectors.

7.3. Different amount of tensile reinforcement (Group A5)

Different amounts of tensile reinforcement are adopted in the hogging moment regions of the continuous composite beams under Group A5. The area of tensile reinforcement of Beam EB2.355-Ap-N1 is 508 [mm.sup.2], while that of Beam EB2.355-Ap-N3 is 1524 [mm.sup.2]. Both the load-deflection curves and development of bending moments of the continuous composite beams under this group are presented in Figure 15. It is found that the amounts of tensile reinforcement do not have significant effect on the load-deflection behaviour of continuous composite beams with practical geometrical features. The load-carrying capacity of Beam EB2.355-Ap-N3 is 6% higher than that of Beam EB2.355-Ap-N1, while the area of tensile reinforcement of Beam EB2.355-Ap-N3 is 3x that of Beam EB2.355-Ap-N1.

Figure 15(b) presents the PMR of the continuous composite beams under this group. It is found that a higher amount of tensile reinforcement in the hogging moment regions typically leads to 3-6% reduction in the PMR due to larger flexural rigidities in the hogging moment regions.

7.4. Different steel strengths (Groups A2 and B2)

7.4.1. Effects of different steel strengths

In order to study the effects of yield strengths of the steel sections on the structural behaviour of continuous composite beams, the analyses in the continuous composite beams under Group A2 are repeated in Group B2 with steel sections with a yield strength at 450 N/[mm.sup.2] Highly non-ductile shear connectors namely, Shear connectors C, are used in the study. The results of analyses of the continuous composite beams under Group B2 are presented in Figure 16. Significant differences in the load-carrying capacities are found due to different steel yield strengths through the comparison on the load deflection curves between Figures 12(a) and 16(a). By increasing the yield strengths from 355 to 450 N/[mm.sup.2], the load-carrying capacities increase by about 20% for most of the continuous composite beams. Moreover, it is noted that, the PMR do not change significantly in composite beams with steel sections of different steel strengths through the comparison between Figures 12(b) and 16(b).

7.4.2. Effects of non-ductile shear connectors

A close examination of Figures 12 and 16 also reveals that the non-ductile deformation characteristics of shear connectors occurs in the medium-size composite beam with a steel strength of 450 N/[mm.sup.2] (Beam EB2.450-Cp-N1) before the formation of the second plastic hinge. This reduces the ultimate load-carrying capacity of the continuous composite beam. However, for the same composite beam with a steel strength of 355 N/[mm.sup.2] (Beam EB2.355-Cp-N1), the non-ductile deformation characteristics of shear connectors occurs right after the formation of the second plastic hinge. Hence, the load-carrying capacity is achieved without premature knock-down. It is noted that, for composite beams with higher steel strengths, larger deformation in the extreme fibres of both the concrete flanges and the steel sections is needed to fully mobilise the cross-sectional moment capacities. This leads to larger slippage at the interfaces between the steel sections and the concrete flanges in the hogging moment regions.

Thus, continuous composite beams with steel sections of steel strengths larger than 450 N/[mm.sup.2] are more prone to suffer from the premature knock-down due to the non-ductile deformation characteristics of shear connectors as opposed to those with steel sections with lower steel strengths.

7.5. Different arrangements of shear connectors with high strength steel (Groups B3 and B4)

In order to study the effects of yield strengths of steel sections on the structural behaviour of continuous composite beams with different arrangements of shear connectors, the analyses under Groups A3 and A4 are repeated under Group B3 and B4 with the adoption of a steel yield strength at 450 N/[mm.sup.2]. Ductile shear connectors are adopted in the continuous composite beams under Group B3, while highly non-ductile shear connectors are adopted in the beams under Group B4.

Both the load-deflection curves obtained from the analyses are presented in Figures 17 and 18 for the continuous composite beams under Groups B3 and B4, respectively. It is found that the continuous composite beams under Groups B3 and B4 behave similar to those under Groups A3 and A4 in both the moment redistribution and the non-ductile deformation characteristics of shear connectors. Higher load-carrying capacities are also obtained in the continuous composite beams under Groups B3 and B4 due to the higher steel strength.

7.6. Different boundary conditions (Groups A0 and C0)

It is noted that, since end-spans are normally considered to be more critical than internal spans in continuous composite beams, all the composite beams under Groups A0 to B4 are double-span continuous composite beams. In order to study the effects of boundary conditions on the structural behaviour of continuous composite beams, the analyses of the composite beams under Group A0 are repeated under Group C0 as continuous composite beams at internal spans. Ductile shear connectors are adopted in the continuous composite beams.

Both the load-deflection curves and the development of bending moments of the continuous composite beams are presented in Figure 19. The comparison between Figures 10 and 19 reveals that the continuous composite beams at internal spans behave quite similar to the continuous composite beams at end-spans with respect to both load-deflection characteristics and development of bending moments. Figures 10(b) and 19(b) present the PMR of the continuous composite beams under Groups A0 and C0. It is found that the PMR of the continuous composite beams at internal spans are around 4% lower than the PMR of the continuous composite beams at end-spans. This is because more hogging moments are concentrated in the internal supports of the continuous composite beams at internal spans than the continuous composite beams at end-spans.

7.7. Distributions of horizontal shear forces and slippage

The occurrence of the non-ductile deformation characteristics of shear connectors can also be observed clearly through examining the distribution of the slippages and the longitudinal forces in the shear connectors of the continuous composite beams. The structural behaviour of Beam EB3.355-Cp-N1 is illustrated in Figure 20, in which the distributions of both bending moments and shear forces along the beam length are plotted at various load levels. The distributions of longitudinal forces and slippage in shear connectors are also plotted.

It is noted that non-ductile deformation characteristics of shear connectors happens in Beam EB3.355Cp-N1 after the formation of the first plastic hinge. It is also found that unloading in the shear connectors spreads from the hogging moment regions where the slippages are normally considered to be high, and propagates towards the sagging moment regions. This leads to fast reduction in the moment capacities in both the hogging and the sagging moment regions, and further leads to overall non-ductile behaviour of the continuous composite beam, which is highly undesirable. As discussed above, continuous composite beams with large section depths (equal to or above 744 mm) and high steel strength (equal to or above 450 N/[mm.sup.2]) are more prone to suffering from the non-ductile deformation characteristics of shear connectors because of the large slippage at the steel-concrete interfaces.

8. Comparison with established design rules

8.1. Moment redistribution

It is noted that for the continuous composite beams with Class 1 steel sections, the maximum percentage of moment redistribution is 25% for cracked section analysis and 40% for uncracked section analysis as stated in Eurocode 4: Part 1.1 (BSI 2004).

Figure 10(b) presents the PMR of the continuous composite beams with different section depths under Groups A0. It should be noted, that by increase the span-over-depth ratios, L/[D.sub.c], from 13.3 to 22.8, the PMR increase from 32 to 49%. The results from codes of practice generally match the result of Beam EB3.355-Ap-N1, whose L/[D.sub.c] ratio is 13.3. However, for the continuous composite beams with larger L/[D.sub.c] ratios, such as Beam EB1.355-Ap-N1, the PMR are much higher than those stated in the relevant codes of practice.

8.2. Ductility of shear connectors

Various codes of practice and design handbooks had their own ways of design for non-ductile shear connectors. Lawson (1990) specified the minimum slippage limit of shear connectors allowed to be used in composite beams. He stated that the shear connectors should be able to retain their shear resistances fully at a slippage of 5 mm, while their shear resistances should not be less than 80% of the maximum values at a slippage of 7 mm. Eurocode 4: Part 1.1 (BSI 2004) specified that the slippage of headed shear connectors in composite beams should be no less than 6 mm. This restricted the types of shear connectors that allowed to be used in composite beams.

Moreover, a minimum in the degree of shear connection was also specified in Eurocode 4: Part 1.1 (BSI 2004) in order to control the maximum slippages in the shear connectors. This ensured ductile deformation characteristics of the shear connectors implicitly. However, it should be noted that the arrangement configurations of shear connectors in composite beams were limited. In the current integrated analysis and design model, the deformation characteristics of shear connectors is pre-defined explicitly in both elastic and plastic stages. Thus, composite beams with a wide range of types and arrangements of shear connectors can be analysed using the proposed analysis and design model.

Moreover, the maximum slippage in shear connectors of the continuous composite beams under Group A0 is also obtained from the numerical results and presented in Table 4. It is found that the maximum slippage of the shear connectors ranges from 0.4 to 4.9 mm for the continuous composite beams to reach Stage 1 failure, while the maximum slippage of the shear connectors ranges from 0.6 to 11.6 mm for the continuous composite beams to reach Stage 2 failure. Normally, the composite beam with a deeper steel section has a larger maximum slippage. Thus, the slippage limit specified in codes of practice and design handbooks (BSI 2004; Lawson 1990) is sufficient for continuous composite beams to reach Stage 1 failure. However, this slippage limit may not be adequate for some continuous composite beams with deep steel sections to reach Stage 2 failure.

8.3. High strength steel

Eurocode 4: Part 1.1 (BSI 2004) specifies the reduction factors for moment capacities of composite beams of high-strength steel (over or equal to 420 N/[mm.sup.2]). Thus, the maximum strain in the critical cross section mobilising the cross-sectional moment capacity can be limited. The reduction factor is determined according to the position of the neutral axis in the composite section. However, it should be noted that the yield strength of the steel section is not considered in that reduction factor.

In the current analysis and design model, the formation of plastic hinges in both hogging and sagging moment regions are determined according to the limiting strain criteria as stated in Equation (2) with the incorporation of the yield strength of the steel section. Thus, the limiting strains of composite sections with high-strength steel are larger when compared with those of composite sections with normal strength steel. This leads to fully developed plastic hinges in composite beams with steel sections of high yield strengths. As observed from the numerical results obtained from finite element analyses of the continuous composite beams under Groups B3 in Figure 19, hogging moments remain constant after the formation of the first plastic hinges. Thus, the hogging moment capacities in the continuous composite beams with high strength steel are considered to be fully mobilised at the formation of plastic hinges at both Stages 1 and 2 failure.

9. Conclusions

A comprehensive parametric study is presented in this Chapter, which studies the structural behaviour of continuous composite beams with different material properties and geometrical configurations. Composite beams with both ductile and non-ductile shear connectors are included to investigate the effects of flexibility of the shear connectors on the overall structural behaviour of continuous composite beams. It is concluded that:

(1) The span-over-depth ratios, L/[D.sub.c], of continuous composite beams affect significantly the PMR. It is found that the limiting PMR given in current codes of practice match the results of continuous composite beams with small to medium span-over-depth ratios (L/[D.sub.c] [less than or equal to] 14). Larger PMR are expected for continuous composite beams with larger span-over-depth ratios. However, it should be noted that the design rules in current codes of practice can still provide conservative prediction towards moment redistribution of continuous composite beams.

(2) The degrees of shear connection in both the hogging and the sagging moment regions have significant effects on the PMR. Since full shear connection can be easily achieved in the hogging moment regions with the normal provision of tensile reinforcement, higher degrees of shear connection in the sagging moment regions normally leads to higher load-carrying capacities of the composite beams, hence higher PMR.

(3) The occurrence of the non-ductile deformation characteristics in shear connectors has significant effects on the structural behaviour of continuous composite beams. Premature knock-down with a sudden deduction in the overall load-carrying capacity over a relative short deflection history and due to the occurrence of non-ductile shear connectors is likely to happen in composite beams with high-steel strengths (equal to or above 450 N/[mm.sup.2]) and large section depth (equal to or above 747 mm). Thus, shear connectors with sufficient ductility should be provided in the design of continuous composite beams with steel sections of high steel strengths and large section depths, especially in the hogging moment regions.

(4) The ductility requirement of shear connectors specified in current codes of practice and design handbooks (BSI 2004; Lawson 1990) is sufficient for continuous composite beams to form the first plastic hinges in the hogging moment regions. However, the codified ductility requirement of shear connectors may not be adequate for some continuous composite beams with deep sections to form the second plastic hinges in the sagging moment regions. In order to ensure that the plastic hinges are formed in the both sagging and hogging moment regions of the continuous composite beams without the occurrence of non-ductile deformation characteristics of shear connectors, the slippage limit of shear connectors should be larger than 12 mm.

http://dx.doi.org/10.1080/13287982.2016.1196572

ARTICLE HISTORY

Received 6 December 2015

Accepted 16 May 2016

Disclosure statement

No potential conflict of interest was reported by the author.

Notes on contributor

Aaron J. Wang is a highly competent engineering director, researcher and civil/structural engineer with more than 15 years of professional and R&D experience. He is currently the director in Project Design and Development with Capital and China with the focus on the technical, R&D and project management on high-rise hybrid commercial buildings. He has authored a total of 30 journal articles, 26 conference papers and 4 technical reports. He is also a frequent speaker in international conferences and seminars and serves as an editorial board member for several internationally established journals. His research interests include steel and composite structures, structural fire engineering and sustainable infrastructures.

References

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Aaron J. Wang

Project Design and Development Management, CapitaLand China Corporate Ltd., Shanghai, P.R. China

CONTACT Aaron J. Wang ([mail]) aaron.wang@capitaland.com

Caption: Figure 1. Finite element model with initial imperfection.

Caption: Figure 2. Material models.

Caption: Figure 3. Typical failure mode of continuous composite beams (Beam CBI).

Caption: Figure 4. Load-deflection curves.

Caption: Figure 5. Load-slippage curves of shear connectors.

Caption: Figure 6. Continous composite beams (End-spans).

Caption: Figure 7. Continous composite beams (Internal spans).

Caption: Figure 8. Moment capacities of design strength at 355 N/[mm.sup.2].

Caption: Figure 9. Moment capacities of design strength at 450 N/[mm.sup.2].

Caption: Figure 10. Numerical results of continuous composite beams under Group A0.

Caption: Figure 11. Numerical results of continuous composite beams under Group A1.

Caption: Figure 12. Numerical results of continuous composite beams under Group A2.

Caption: Figure 13. Numerical results of continuous composite beams under Group A3.

Caption: Figure 14. Numerical results of continuous composite beams under Group A4.

Caption: Figure 15. Numerical results of continuous composite beams under Group A5.

Caption: Figure 16. Numerical results of continuous composite beams under Group B2.

Caption: Figure 17. Numerical results of continuous composite beams under Group B3.

Caption: Figure 18. Numerical results of continuous composite beams under Group B4.

Caption: Figure 19. Numerical results of continuous composite beams under Group C0.

Caption: Figure 20a. Structural behaviour of Beam EB3.355-Cp-N1 (first crack).

Caption: Figure 20b. Structural behaviour of Beam EB3.355-Cp-N1 (Stage 1 failure).

Caption: Figure 20c. Structural behaviour of Beam EB3.355-Cp-N1 (Stage 2 failure).
Table 1. Summary of tests on continuous composite beams.

Researcher      Year   Test    Span, L (mm)     Loading    Depth of
                                                  and       steel
                                               boundary    section,
                                               condition    D (mm)

Ansourian       1981   CTB1     4000+5000                    200
(1981)                 CTB3     4500+4500                    190

Hope Gill       1976   cB11   2240+4480+2240                 207
and Johnson
(1976)                 cB12   5200+5800+5200                 207

Hamada          1976   cB1      3658+3658                    313
and Longworth
(1976)

Researcher      Depth of    Width of    [B.sub.s]/L   L/[D.sub.c]
                composite   concrete
                section,     flange,
                [D.sub.c]   [B.sub.s]
                  (mm)        (mm)

Ansourian          300         800         0.18          15.0
(1981)             290        1300         0.29          15.5

Hope Gill          283         915         0.20          15.9
and Johnson
(1976)             309        1220         0.21          18.7

Hamada             415        1219         0.33           8.8
and Longworth
(1976)

Notes: All shear connectors are 19-mm-headed shear connectors. All
slabs are solid.

Table 2. Load-carrying capacities at Stages 1 and 2 failure.

                           Finite element model

                             Stage 1 failure

Test   [P.sub.Test]   Failure location   [P.sub.FEM,1]
           (kN)                              (kN)

CTB1      204.3       Mid-span               192.4
CTB3      575.2       Internal support       503.4
CB11      291.2       Mid-span               209.3
CB12      835.0       Internal support       581.3
CB1       1199.0      Internal support       961.3

                            Finite element model

                              Stage 2 failure

Test   [P.sub.Test]   Failure            [P.sub.MFE,2]
           (kN)       location               (kN)

CTB1      204.3       Internal support       210.1
CTB3      575.2       Mid-span               572.3
CB11      291.2       Internal support       246.7
CB12      835.0       Mid-span               773.1
CB1       1199.0      Mid-span              1178.6

                       Finite element
                            model

                       Stage 2 failure

Test   [P.sub.Test]   [[epsilon].sub.t]
           (kN)              (%)

CTB1      204.3             0.69
CTB3      575.2             0.85
CB11      291.2             0.64
CB12      835.0             1.10
CB1       1199.0            0.48

Notes: [[epsilon].sub.t] represents strains of tensile
reinforcement at failure.

Table 3. Details of parametric study on continuous composite beams.

End-span/       Yield        Group   Label            Type of
internal       strength                                shear
span           of steel                              connector
               section,
              [p.sub.y]
            (N/[mm.sub.2])

End-span         355          A0     EB1.355-Ap-N1       A
                                     EB2.355-Ap-N1
                                     EB3.355-Ap-N1
                              A1     EB1.355-Bp-N1       B
                                     EB2.355-Bp-N1
                                     EB3.355-Bp-N1
                              A2     EB1.355-cp-N1       C
                                     EB2.355-cp-N1
                                     EB3.355-cp-N1
                              A3     EB2.355-Ap-N1       A
                                     EB2.355-Ae-N1
                                     EB2.355-Aa-N1
                              A4     EB2.355-cp-N1       C
                                     EB2.355-ce-N1
                                     EB2.355-ca-N1
                              A5     EB2.355-Ap-N1       A
                                     EB2.355-Ap-N2
                                     EB2.355-Ap-N3
End-span         450          B2     EB1.450-cp-N1       C
                                     EB2.450-cp-N1
                                     EB3.450-cp-N1
                              B3     EB2.450-Ap-N1       A
                                     EB2.450-Ae-N1
                                     EB2.450-Aa-N1
                              B4     EB2.450-cp-N1       C
                                     EB2.450-ce-N1
                                     EB2.450-ca-N1
internal         355          C0     IB1.355-Ap-N1       A
span
                                     IB2.355-Ap-N1
                                     IB3.355-Ap-N1

End-span/     Number        Area of       Depth of    L/[D.sub.c]
internal     of shear       tensile       composite
span        connectors   reinforcement,   section,
            per trough     [A.sub.t]      [D.sub.c]
                          ([mm.sup.2])      (mm)

End-span        2             508           438.9        22.8
                                            595.1        16.8
                                            747.5        13.3
                2             508           438.9        22.8
                                            595.1        16.8
                                            747.5        13.3
                2             508           438.9        22.8
                                            595.1        16.8
                                            747.5        13.3
                2             508           595.1        16.8
                1
               0.5
                2             508           595.1        16.8
                1
               0.5
                2             508           595.1        16.8
                              1016
                              1524
End-span        2             508           438.9        22.8
                                            595.1        16.8
                                            747.5        13.3
                2             508           595.1        16.8
                1
               0.5
                2             508           595.1        16.8
                1
               0.5
internal        2             508           438.9        22.8
span
                                            595.1        16.8
                                            747.5        13.3

End-span/     Degree of shear connection
internal
span           Sagging         Hogging
               moment,         moment,
            [[eta].sub.s]   [[eta].sub.h]

End-span        1.80            4.32
                0.62            4.32
                0.36            4.32
                1.80            4.32
                0.62            4.32
                0.36            4.32
                1.80            4.32
                0.62            4.32
                0.36            4.32
                0.62            4.32
                0.31            2.16
                0.15            0.92
                0.62            4.32
                0.31            2.16
                0.15            0.92
                0.62            4.32
                0.62            2.16
                0.62            1.44
End-span        1.42            4.32
                0.49            4.32
                0.28            4.32
                0.49            4.32
                0.24            2.16
                0.12            0.92
                0.49            4.32
                0.24            2.16
                0.12            0.92
internal        1.49            4.32
span
                0.51            4.32
                0.30            4.32

Notes: The span of all beams is 10 m.

Table 4. Summary of numerical results on continuous composite beams
under Groups A0, A1 and A2.

                          Sagging     [M.sub.sag]'/     Hogging
                          moment       [M.sub.sag]      moment
Group   Label            capacity,                     capacity,
                        [M.sub.sag]                   [M.sub.hog]
                           (kNm)                         (kNm)

A0      EB1.355-Ap-N1      342.5          1.00           186.7
        EB2.355-Ap-N1     1105.6          1.00           657.2
        EB3.355-Ap-N1     2658.0          1.00          1890.3
A1      EB1.355-Bp-N1      342.5          1.00           186.7
        EB2.355-Bp-N1     1105.6          1.00           657.2
        EB3.355-Bp-N1     2258.9          0.84          1823.2
A2      EB1.355-Cp-N1      342.5          1.00           186.7
        EB2.355-cp-N1     1105.6          1.00           657.2
        EB3.355-cp-N1     2056.9          0.77          1795.6

                        [M.sub.hog]'/   Ultimate
                         [M.sub.hog]      load-
Group   Label                           carrying     W'/W
                                        capacity,
                                         W (kN)

A0      EB1.355-Ap-N1       1.00          745.8      1.00
        EB2.355-Ap-N1       1.00         2253.1      1.00
        EB3.355-Ap-N1       1.00         5632.1      1.00
A1      EB1.355-Bp-N1       1.00          745.8      1.00
        EB2.355-Bp-N1       1.00         2253.1      1.00
        EB3.355-Bp-N1       0.96         5005.2      0.88
A2      EB1.355-Cp-N1       1.00          745.8      1.00
        EB2.355-cp-N1       1.00         2253.1      1.00
        EB3.355-cp-N1       0.95         4506.4      0.80

                          Percentage          Maximum
                          of moment          slippage,
Group   Label           redistribution    [S.sub.max] (mm)
                             (%)
                                         Stage 1   Stage 2

A0      EB1.355-Ap-N1         49           0.4       0.6
        EB2.355-Ap-N1         41           3.1       4.9
        EB3.355-Ap-N1         32           4.9      11.6
A1      EB1.355-Bp-N1         49           0.4       0.6
        EB2.355-Bp-N1         41           3.1       4.9
        EB3.355-Bp-N1         32           4.9      14.2
A2      EB1.355-Cp-N1         49           0.4       0.6
        EB2.355-cp-N1         41           3.1       4.9
        EB3.355-cp-N1         32           4.9      18.9

Notes: [M.sub.sag]' represents the sagging moment capacity with
ductile shear connectors. [M.sub.hog]' represents the hogging moment
capacity with ductile shear connectors. W represents the load-
carrying capacity with ductile shear connectors. Alt data are
obtained at Stage 2 failure.
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